Aleksandar Ivić
Mean square estimates for $Z_2(s) = \int_1^\infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are discussed, and some related Mellin transforms of quantities connected with the fourth power moment of $|ζ(1/2+ix)|^4$.
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In this note we show that for an arbitrary semisimple Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles.
It is shown that Mellin transforms of p-adic Whittaker functions exist for generic characters. For good choices of vectors they are rational functions. For class one vectors they can be calculated explicitly. It turns out that they are automorphic L-factors times normalization factors.
In this paper, we present a simple analytic proof of Siegel's theorem that concerns the lower bound of $L(1,χ)$ for primitive quadratic $χ$. Our new method compares an elementary lower bound with an analytic upper bound obtained by the inverse Mellin transform of $Γ(s)$.
Some relations involving the Mellin and Laplace transforms of powers of the classical Hardy function $$ Z(t) := ζ(1/2+it)\bigl(χ(1/2+it)\bigr)^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s) $$ are obtained. In particular, we discuss some mean square identities and their consequences.
The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |ζ(1/2+ix)|^{2k}x^{-s}{\rm d} x (k = 1,2,...)$ is investigated. Analytic continuation and mean square estimates of ${\cal Z}_k(s) $ are discussed, as well as connections with power moments of $|ζ(1/2+ix)|$, with the special emphasis on the cases $k = 1,2$.
The aim of our present work here is to present few results in the theory of Mellin transforms using the method that S. Ramanujan used in proving his Master Theorem. Further applications of our results for some number-theoretic functions such as the prime counting function and the zeta function are established.
Various properties of the Mellin transform function $$ {\cal M}_k(s) := \int_1^\infty Z^k(x)x^{-s}dx $$ are investigated, where $$ Z(t) := ζ(1/2+it){\bigl(χ(1/2+it)\bigr)}^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s) $$ is Hardy's function and $ζ(s)$ is Riemann's zeta-function. Connections with power moments of $|ζ(1/2+it)|$ are established, and natural boundaries of ${\cal M}_k(s)$ are discussed.
We say that a function f defined on R or Qp has a well defined weak Mellin transform (or weak zeta integral) if there exists some function $M\_f(s)$ so that we have $Mell(ϕ\star f,s) = Mell(ϕ,s)M\_f(s)$ for all test functions $ϕ$ in $C\_c^\infty(R^*)$ or $C\_c^\infty(Q\_p^*)$. We show that if $f$ is a non degenerate second degree character on R or Qp, as defined by Weil, then the weak Mellin transform of $f$ satisfies a functional equation and cancels only for $\Re(s) = 1/2$. We then show that if $f$ is a non degenerate second degree character defined on the adele ring $A\_Q$, the same stateme…
This paper gives a survey of known results concerning the Laplace transform $$ L_k(s) := \int_0^\infty |ζ(1/2+ ix)|^{2k}{\rm e}^{-sx}{\rm d} x \qquad(k \in N, \R s > 0), $$ and the (modified) Mellin transform $$ {\cal Z}_k(s) := \int_1^\infty|ζ(1/2+ ix)|^{2k}x^{-s}{\rm d} x\qquad(k\in N), $$ where the integral is absolutely convergent for $\R s \ge c(k) > 1$. Also some new results on these integral transforms of $|ζ(1/2+ ix)|^{2k}$ are given, which have important connections with power moments of the Riemann zeta-function $ζ(s)$.
Let ${\cal Z}_1(s) = \int_1^\infty |ζ({1\over2}+ix)|^2x^{-s}{\rm d}x (σ= \Re s > 1)$. A result concerning analytic continuation of ${\cal Z}_1(s)$ to $\bf C$ is proved, and also a result relating the order of ${\cal Z}_1(σ+ it) (1/2 \le σ\le 1, t\ge t_0)$ to the order of ${\cal Z}_1({1\over2}+it)$.
Let $f$ a smooth function with compact support defined on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$, we prove a formula for the Mellin transform of $f$, then we can define the micro-local lift $dε_{it}$ to $SL(2,\mathbb{C})$. We calculate $(f,dε_{it})$ for $f$ a cuspidal form and for $f$ an incomplete Eisenstein series. We also establish asymptotic estimates when $t$ tends to $\infty$. We conjecture that a new positive distribution $dε_{it}^F$, constructed with the Friedrichs' symmetrization technique, satisfies the same asymptotic estimates that $dε_{it}$. This would imply the quant…
A key theorem formulated in the context of functional Mellin transforms generalizes the important relationship $\exp\mathrm{tr} M=\det\exp M$. Along with the involution symmetry of the zeta function, the theorem suggests a strategy for tackling the Riemann hypothesis.
Mellin transforms, transfinite diameter and rational approximations of integrals
本文提出一个高维无理性准则，用于证明由Mellin积分表示的周期的无理性，通过超越直径上界分析多变量Vandermonde矩阵。
We present a summation rule using the Mellin transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the negative integers are expressed in terms of Bernoulli polynomials. We also show identities involving exponential and Hermite polynomials.
A Motivic Pairing and the Mellin Transform in Function Fields
本文定义了关联abelian Anderson A-模的A-动形与对偶A-动形的两个配对，通过其特殊化给出指数与对数函数的精确公式及无穷乘积表示，并以此证明了Carlitz zeta值的Mellin变换公式类似物。
Mellin Transform Formulas for Anderson Modules and Hints of Modularity
本文引入阿贝尔可一致化安德森模的对数函数公式，结合动机映射关联德林费尔德模的对偶戈斯L函数特殊值与t-余模的刚性解析平凡化，还将L值与矩阵值模形式特殊值关联，推广了相关向量值德林费尔德模形式。
This paper continues the study of the structures induced on the ``invisible boundary'' of the modular tower and extends some results of math.NT/0102006. We start with a systematic formalism of pseudo-measures generalizing the well-known theory of modular symbols for SL(2). These pseudo-measures, and the related integral formula which we call the Levy-Mellin transform, can be considered as an ``infinity-adic'' version of Mazur's p-adic measures introduced in the seventies in the theory of p-adic interpolation of Mellin transforms of cusp forms. A formalism of iterated Levy-Mellin transform in t…
The Mellin's transforms of $\dfrac{1}{\operatorname{arctanh} x}$ and $\dfrac{1}{\sqrt{1-x^2}\,\operatorname{arctanh} x}$
本文通过复变积分技术，推导了$1/\operatorname{arctanh} x$和$1/(\sqrt{1-x^{2}}\,\operatorname{arctanh} x)$的 Mellin 变换的显式闭式表达式，扩展了相关算术性质的研究。
The relationship between the Ohno relation and multiple polylogarithms are discussed. Using this relationship, the algebraic reduction of the Ohno relation is given.
Mellin transform formulas for Drinfeld modules
本研究借助新框架推导Drinfeld模的对数公式，将其表为t-动机刚性解析平凡化乘积的动机映射取值，特殊化得到Goss L-函数特殊值表达式，并推广至带Tate扭的Drinfeld模，为特征p域提供了Hasse-Weil型ζ函数积分表示的类似物。
Let $p$ be a prime number and $\mathbb{C}_p$ the completion of algebraic closure of $\mathbb{Q}_p$. Let $K$ be an algebraic number field. We fix an embedding $ι_p:\overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ and denote $K_p$ the completion of $K$ with respect to the embedding $ι_p$. Let $g(z)\in K[[z]]$ and denote by $\mathcal{M}(g)(z)\in \tfrac{1}{z}K\left[\left[\tfrac{1}{z}\right]\right]$ the formal Mellin transform of $g(z)$. In this article, we prove that if $\mathcal{M}(g)(z)$ has a good Padé approximation, the special values $\mathcal{M}(g)(α)$ are convergent in $K_p$ and irrationa…
Motivic Mellin transforms
本文将Mellin变换引入motivic积分领域，构建了在Mellin和Fourier变换下稳定的新函数类，提供Fubini定理与变量替换公式，并推广了前人理论与转移原理。
Let L(s) = L(s, π) be the standard L-function of a cuspidal representation πof GL(m,A) where A denotes the adèles of the field of rationals. We consider the integral, on the real line Re(s)= 1/2, of the squared absolute value of L(s)/s. In an earlier paper, partly with P. Sarnak (arxiv:2203.12475) we obtained a universal lower bound on this integral, independently of m. In this paper, for m fixed, we first obtain a universal lower bound for the integral on an interval [-A logC, A log C] where C is the analytic conductor of π; this bound is of order c(log C)^{-1/2} ; A, c are absolute positive …
At the tree level, the maximally helicity violating amplitudes of N gauge bosons in open superstring theory and of N gravitons in supergravity are known to have simple representations in terms of tree graphs. For superstrings, the graphs encode integral representations of certain generalized Gaussian hypergeometric functions of kinematic invariants while for supergravity, they represent specific kinematic expressions constructed from spinor-helicity variables. We establish a superstring/supergravity correspondence for this class of amplitudes, by constructing a mapping between the positions of…
On $p$-adic incomplete Mellin transforms and $p$-adic incomplete gamma-functions
本文提出了一种在更弱条件下构造 $p$-adic 不完整 gamma 函数的方法，利用 $p$-adic 积分变换，适用于所有除了有限多个素数外的素数。
For a complex variable $s$ and real parameters $a$ and $λ$ with $a>0$, let $ϕ(s,a,λ)$ denote the Lerch zeta-function with a complex variable, $ϕ^{\ast}(s,a,λ)$ a slight modification of $ϕ(s,a,λ)$ defined by extracting the (possible) singularity of $ϕ(s,a,λ)$ at $s=1$, and $(ϕ^{\ast})^{(m)}(s,a,λ)$ for any $m\in\mathbb{Z}$ the $m$th derivative with respect to $s$ if $m\geq0$, while if $m\leq0$ the $|m|$-th primitive defined with its initial point at $s+\infty$. The present paper aims to study asymptotic aspects of $(ϕ^{\ast})^{(m)}(s,a,λ)$, transformed through the Laplace-Mellin and Riemann-Lio…
We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line Re $s=1/2$. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. These polynomials possess the functional equation $p_n(s)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s)$. Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part function…
We consider the Mellin transforms of certain Chebyshev functions based upon the Chebyshev polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line or on the real line. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend this result to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;β)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s;β)$. We then present the generalization to the Mell…
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz number…
It is a well-known fact that Riemann Hypothesis will follows if the function identically equal to -1 can be arbitrarily approximated in the norm $\norma{.}$ of $L^{2}([0,1],dx)$ by functions of the form $f(x)=\sum_{k=1}^{n}a_{k} ρ(\frac{θ_{k}}{x})$, where $ρ(x)\adef\pfrac{x}$, and $a_{k}\in\cc$, $0<θ_{k}\leqslant 1$ satisfies $\sum_{k=1}^{n}a_{k} θ_{k}=0$. Parsevall Identity $\norma{f(x)+1}^{2}=\sum_{n\in\zz}\modulo{c(n)}^{2}$ is a possible tool to compute or estimate this norm. In this note we give an expression for the Fourier coefficients $c(n)$ of $f+1$, when $f$ is a function defined as a…
The Parabolic Mellin Transform: Gamma and Zeta Integral Representations
本文介绍了抛物线梅林变换（PMT），并通过其对高斯核的处理，得到经典汉克尔表示的抛物线轮廓形式，并利用该变换推导了Hurwitz泽塔函数、黎曼泽塔函数和狄利克雷eta函数的积分表示，同时提供了Gamma型和泽塔型特殊函数的统一变换字典，并将黎曼假设和林德洛夫假设重新表述为抛物线轮廓积分零点和增长的等价形式。
In this paper, we evaluate archimedean zeta integrals for automorphic $L$-functions on $GL_n \times GL_{n-1+\ell}$ and on $ SO_{2n+1} \times GL_{n+\ell}$, for $\ell=-1$, $0$, and $1$. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals.
Let $G$ be the $ax+b$ group. There are essentially two irreducible infinite dimensional unitary representations of $G$, $(μ, L^2(\mathbb R^+))$ and $(μ^*, L^2(\mathbb R^+))$. In this paper, we give various characterizations about smooth vectors of $μ$ and their Mellin transforms. Let $\f d$ be a linear sum of delta distributions supported on the the positive integers $\mathbb Z^+$. We study the Mellin transform of the matrix coefficients $μ_{ \f d, f}(a)$ with $f$ smooth. We express these Mellin transforms in terms of the Dirichlet series $L(s, \f d)$. We determine a sufficient condition such …
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